| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > meaiuninc3 | Structured version Visualization version GIF version | ||
| Description: Measures are continuous from below: if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is the general case of Proposition 112C (e) of [Fremlin1] p. 16 . This theorem generalizes meaiuninc 46496 and meaiuninc2 46497 where the sequence is required to be bounded. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
| Ref | Expression |
|---|---|
| meaiuninc3.p | ⊢ Ⅎ𝑛𝜑 |
| meaiuninc3.f | ⊢ Ⅎ𝑛𝐸 |
| meaiuninc3.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
| meaiuninc3.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| meaiuninc3.z | ⊢ 𝑍 = (ℤ≥‘𝑁) |
| meaiuninc3.e | ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) |
| meaiuninc3.i | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) |
| meaiuninc3.s | ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
| Ref | Expression |
|---|---|
| meaiuninc3 | ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | meaiuninc3.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
| 2 | meaiuninc3.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
| 3 | meaiuninc3.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑁) | |
| 4 | meaiuninc3.e | . . 3 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) | |
| 5 | meaiuninc3.p | . . . . . 6 ⊢ Ⅎ𝑛𝜑 | |
| 6 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑛 𝑘 ∈ 𝑍 | |
| 7 | 5, 6 | nfan 1899 | . . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑘 ∈ 𝑍) |
| 8 | meaiuninc3.f | . . . . . . 7 ⊢ Ⅎ𝑛𝐸 | |
| 9 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑛𝑘 | |
| 10 | 8, 9 | nffv 6916 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘𝑘) |
| 11 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑛(𝑘 + 1) | |
| 12 | 8, 11 | nffv 6916 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘(𝑘 + 1)) |
| 13 | 10, 12 | nfss 3976 | . . . . 5 ⊢ Ⅎ𝑛(𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)) |
| 14 | 7, 13 | nfim 1896 | . . . 4 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
| 15 | eleq1w 2824 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍)) | |
| 16 | 15 | anbi2d 630 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍))) |
| 17 | fveq2 6906 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘)) | |
| 18 | fvoveq1 7454 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1))) | |
| 19 | 17, 18 | sseq12d 4017 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))) |
| 20 | 16, 19 | imbi12d 344 | . . . 4 ⊢ (𝑛 = 𝑘 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))))) |
| 21 | meaiuninc3.i | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) | |
| 22 | 14, 20, 21 | chvarfv 2240 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
| 23 | meaiuninc3.s | . . . 4 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) | |
| 24 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘𝑀 | |
| 25 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘(𝐸‘𝑛) | |
| 26 | 24, 25 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) |
| 27 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑛𝑀 | |
| 28 | 27, 10 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) |
| 29 | 2fveq3 6911 | . . . . 5 ⊢ (𝑛 = 𝑘 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑘))) | |
| 30 | 26, 28, 29 | cbvmpt 5253 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘))) |
| 31 | 23, 30 | eqtri 2765 | . . 3 ⊢ 𝑆 = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘))) |
| 32 | 1, 2, 3, 4, 22, 31 | meaiuninc3v 46499 | . 2 ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘))) |
| 33 | fveq2 6906 | . . . 4 ⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛)) | |
| 34 | 10, 25, 33 | cbviun 5036 | . . 3 ⊢ ∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) |
| 35 | 34 | fveq2i 6909 | . 2 ⊢ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘)) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
| 36 | 32, 35 | breqtrdi 5184 | 1 ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ⊆ wss 3951 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1c1 11156 + caddc 11158 ℤcz 12613 ℤ≥cuz 12878 ~~>*clsxlim 45833 Meascmea 46464 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-lm 23237 df-xms 24330 df-ms 24331 df-xlim 45834 df-salg 46324 df-sumge0 46378 df-mea 46465 |
| This theorem is referenced by: (None) |
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